Theorem sto1i

Description: The state of a subspace plus the state of its orthocomplement. (Contributed by NM, 24-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypothesis	sto1.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	sto1i	$$⊢S\in \mathrm{States}\to S\left(A\right)+S\left(\perp \left(A\right)\right)=1$$

Proof

Step	Hyp	Ref	Expression
1		sto1.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2	1	chjoi	$$⊢A{\vee }_{\mathscr{H}}\perp \left(A\right)=\mathscr{H}$$
3	2	fveq2i	$$⊢S\left(A{\vee }_{\mathscr{H}}\perp \left(A\right)\right)=S\left(\mathscr{H}\right)$$
4	1	choccli	$$⊢\perp \left(A\right)\in {\mathbf{C}}_{\mathscr{H}}$$
5	1 4	pm3.2i	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge \perp \left(A\right)\in {\mathbf{C}}_{\mathscr{H}}\right)$$
6	1	chshii	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}$$
7		shococss	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}\to A\subseteq \perp \left(\perp \left(A\right)\right)$$
8	6 7	ax-mp	$$⊢A\subseteq \perp \left(\perp \left(A\right)\right)$$
9		stj	$$⊢S\in \mathrm{States}\to \left(\left(\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge \perp \left(A\right)\in {\mathbf{C}}_{\mathscr{H}}\right)\wedge A\subseteq \perp \left(\perp \left(A\right)\right)\right)\to S\left(A{\vee }_{\mathscr{H}}\perp \left(A\right)\right)=S\left(A\right)+S\left(\perp \left(A\right)\right)\right)$$
10	5 8 9	mp2ani	$$⊢S\in \mathrm{States}\to S\left(A{\vee }_{\mathscr{H}}\perp \left(A\right)\right)=S\left(A\right)+S\left(\perp \left(A\right)\right)$$
11		sthil	$$⊢S\in \mathrm{States}\to S\left(\mathscr{H}\right)=1$$
12	3 10 11	3eqtr3a	$$⊢S\in \mathrm{States}\to S\left(A\right)+S\left(\perp \left(A\right)\right)=1$$